Many commenters have expressed the opinion that the net radiation at the earth’s surface is quite small and therefore radiation doesn’t play a big part in establishing the temperature of the lower atmosphere. The lower atmosphere, for reference, is usually known as the troposphere.
Here is an often misunderstood reference diagram, from Trenberth and Kiehl (1997):
As a digression, this approach to estimating the energy balance at the top of atmosphere and the surface isn’t something new. There is probably a 100 year history of different investigators trying to calculate these values..
Let’s take a look at this diagram of global annual average estimates.
What is the surface energy “budget”?
- Solar radiation absorbed = 168 W/m²
- Atmospheric radiation absorbed = 324 W/m²
- Total radiation received = 492 W/m²
- Total radiation emitted = 390 W/m²
- Net radiation = -102 W/m²
The balance of energy from the surface (102 W/m²) is moved by convection and latent heat.
As a different perspective, some people look at the problem as the balance of longwave radiation – in which case the surface radiates 390 W/m² (as before) and receives atmospheric radiation (aka “back radiation” or DLR) of 324 W/m².
From this perspective net longwave radiation is 66 W/m².
What does this mean? If the net value is low does that mean that radiation is unimportant?
A World with No Convection
As we often do on this blog, we will consider an unreal world in an attempt to explain something otherwise hidden..
Let’s suppose that our atmosphere didn’t support convection/conduction from the surface and the lower atmosphere didn’t have any convection at all.
Therefore, the only mechanism for heat transfer is radiation. What would this world be like?
It would be challenging to do the full calculations of radiative transfer considering tens of thousands of absorption lines, but a handy approximation is the grey atmospheric model. In this model the atmosphere absorbs uniformly across all long wavelengths (and is transparent to solar radiation).
The grey atmosphere makes the maths a lot simpler, and, as usual the maths is saved for a section at the end for those interested.
You can see the equations of radiative transfer in CO2 – An Insignificant Trace Gas? Part Three. In essence, the proportion of radiation absorbed in a small section of the atmosphere is proportional to the density of the “radiative absorbers” in the atmosphere.
I picked an arbitrary value of absorption, resulting in this temperature profile:
It’s not particularly important for this analysis but the temperature of the ground would be higher than the temperature in the atmosphere just above the ground. In a world with convection and conduction this couldn’t happen, but in a world with only radiation to exchange heat between the surface and atmosphere it would.
The “optical depth” is a key property in this approach:
By convention, optical depth, χ, is measured from the top of the atmosphere and is a way of expressing the total absorption through the atmosphere.
What about the radiation up and down from the surface to the top of atmosphere? First we will look at it expressed against optical depth:
And now against height:
As you can see, at the top of atmosphere the downward longwave radiation is zero. That’s because there’s no atmosphere to radiate from.
And also at the top of atmosphere, the upward longwave radiation = 240 W/m². This balances the solar (shortwave) radiation as the climate in this world is in overall equilibrium.
Now, let’s look at the picture of up and down fluxes when solar radiation (of 240W/m²) is included:
At each height the total up and down fluxes are balanced. The net radiation at each height is zero.
So clearly in this unusual world radiation has no effect..
Except that without the absorption and emission at each height in the atmosphere the surface temperature would be a lot lower. If we increase the optical thickness of the atmosphere the surface temperature increases. And yet still net radiation is zero.
Perversely, in the real world, convection acts to reduce temperatures, because it redistributes heat more effectively than radiation. As convection takes more and more heat, the net radiation at the surface becomes larger.
When net radiation is zero that means radiation is doing everything.
As net radiation moves away from zero, then either other heat transfer mechanisms are in place, or that part of the atmosphere/surface is heating or cooling.
The equation for radiative transfer through a layer of atmosphere (often known as Schwarzschild’s equation):
dI = -Ikρ.dz + Bkρ.dz
where I = radiation intensity, k = absorption coefficient, ρ = density of absorber, B = blackbody emission, dz = incremental height
dI/dχ = I – B
where optical density, χ = kρ.dz
We want to apply the equation above for a plane parallel atmosphere. A lot of maths shows that to a good approximation I may be replaced by F (flux), and B by πB (blackbody function integrated over a hemisphere) if χ is replaced by χ* = χ.1.66. This is known as the diffusivity approximation.
Considering the up and down fluxes:
dF↑/dχ* = F↑ – πB
-dF↓/dχ* = F↓ - πB
And considering that the change in F↑ - F↓ with height (or optical thickness) must be zero in equilibrium – otherwise that layer of the atmosphere would be heating or cooling.
We find that:
B = (χ*+1). F0/2π
where F0 = average absorbed solar radiation at the surface, and because πB = σT4 we can calculate T as a function of χ* and therefore as a function of height. Also the up and down fluxes are linear functions of optical thickness:
F↑ = (χ*+2). F0/2
F↓ = χ*. F0/2